Mathematical Methods of Game and Economic Theory Book

Preface to the Dover Edition     iii
Preface (1982)     vii
Summary of Results: A Guideline for the Reader     xxi
Contents of Other Possible Courses     xxvii
Notations     xxix
Optimization and Convex Analysis     1
Minimization Problems and Convexity     3
Strategy sets and loss functions     4
Optimization problem     4
Allocation of available commodities     5
Resource and service operators     6
Extension of loss functions     8
Sections and epigraphs     10
Decomposition principle     11
Product of a loss function by a linear operator     11
Example: Inf-convolution of functions     12
Decomposition principle     13
Another decomposition principle     15
Mixed strategies and convexity     17
Motivation: extension of strategy sets and loss functions     18
Mixed strategies and linearized loss functions     19
Interpretation of mixed strategies     21
Case of finite strategy sets     21
Representation by infinite sequences of pure strategies     22
Linearized extension of maps and the barycentric operator     24
Interpretation of convex functions in terms of risk aversion     25
Elementary properties of convex subsets and functions     25
Indicators, support functions and gauges     27
Indicators and support functions     28
Reformulation of the Hahn-Banach theorem     31
The bipolar theorem     32
Recession cones and barrier cones     34
Interpretation: production sets and profit functions     35
Gauges     38
Existence, Uniqueness and Stability of Optimal Solutions     42
Existence and uniqueness of an optimal solution     43
Structure of the optimal set     43
Existence of an optimal solution     45
Continuity versus compactness     45
Lower semi-continuity of convex functions in infinite dimensional spaces     45
Fundamental property of lower semi-continuous and compact functions     46
Uniqueness of an optimal solution     47
Non-satiation property     48
Minimization of quadratic functionals on convex sets     48
Hilbert spaces     49
Existence and uniqueness of the minimal solution     49
Characterization of the minimal solution     50
Projectors of best approximation      51
The duality map from an Hilbert space onto its dual     52
Minimization of quadratic functionals on subspaces     54
The fundamental formula     54
Orthogonal right inverse     56
Orthogonal left inverse     57
Another decomposition property     58
Interpretation     59
Perturbation by linear forms: conjugate functions     60
Conjugate functions     60
Characterization of lower semi-continuous convex functions     61
Examples of conjugate functions     62
Elementary properties of conjugate functions     64
Interpretation: cost and profit functions     65
Stability properties: an introduction to correspondences     66
Upper semi-continuous correspondences     66
Lower semi-continuous correspondences     68
Closed correspondences     70
Construction of upper semi-continuous correspondences     73
Compactness and Continuity Properties     75
Lower semi-compact functions     76
Coercive and semi-coercive functions     76
Functions such that f* is continuous at 0     77
Lower semi-compactness of linear forms     78
Constraint qualification hypothesis     79
Case of infinite dimensional spaces     81
Extension to compact subsets of mixed strategies     82
Proper maps and preimages of compact subsets     83
Proper maps     84
Compactness of some strategy sets     85
Examples where the map L* + 1 is proper     88
Continuous convex functions     90
A characterization of lower semi-continuous convex functions     90
A characterization of continuous convex functions     91
Examples of continuous convex functions     93
Continuity of gL and Lf     94
Continuous convex functions (continuation)     95
Strong continuity of lower semi-continuous convex functions     96
Estimates of lower semi-continuous convex functions     97
Characterization of continuous convex functions     98
Continuity of support functions     99
Maximum of a convex function: extremal points     100
Differentiability and Subdifferentiability: Characterization of Optimal Solutions     103
Subdifferentiability     105
Definitions     105
Examples of subdifferentials     106
Subdifferentiability of continuous convex functions     108
Upper semi-continuity of the subdifferential     109
Characterization of subdifferentiable convex functions     110
Differentiability and variational inequalities     111
Definitions     111
Differentiability and subdifferentiability     112
Legendre transform     113
Interpretation: marginal profit     114
Variational inequalities     114
Differentiability from the right     115
Definition and main inequalities     115
Derivatives from the right and the support function of the subdifferential     117
Derivative of a pointwise supremum     118
Local [epsilon]-subdifferentiability and perturbed minimization problems     120
Approximate optimal solutions in Banach spaces     121
The approximate variational principle     123
Local [epsilon]-subdifferentiability     124
Perturbation of minimization problems     126
Proof of Ekeland-Lebourg's theorem     130
Introduction to Duality Theory     133
Dual problem and Lagrange multipliers     135
Lagrangian     136
Lagrange multipliers and dual problem     137
Marginal interpretation of Lagrange multipliers     139
Example      140
Case of linear constraints: extremality relations     142
Generalized minimization problem     143
Extremality relations     145
The fundamental formula     146
Minimization problem under linear constraints     148
Minimization of a quadratic functional under linear constraints     148
Minimization problem under linear equality constraints     149
Duality and the decomposition principle     150
The decentralization principle     151
Conjugate function of gL     152
Conjugate function of f[subscript 1]+f[subscript 2]     153
Minimization of the projection of a function     154
Minimization on the diagonal of a product     154
Existence of Lagrange multipliers in the case of a finite number of constraints     155
The Fenchel existence theorem     156
Stability properties     157
Applications to subdifferentiability     158
Case of nonlinear constraints: The Uzawa existence theorem     159
Game Theory and the Walras Model of Allocation of Resources     363
Two-Person Games: An Introduction     165
Some solution concepts     167
Description of the game     167
Shadow minimum     367
Conservative solutions and values     168
Non-cooperative equilibrium     169
Pareto minimum     170
Core of a two-person game     171
Selection of strategy of the core     171
Examples: some finite games     172
Example     173
Coordination game     175
Prisoner's dilemma     178
Game of chicken     180
The battle of the sexes     182
Example: Analysis of duopoly     183
The model of a duopoly     184
The set of Pareto minima     185
Conservative solutions     185
Non-cooperative equilibria     186
Stackelberg equilibria     187
Stackelberg disequilibrium     187
Example: Edgeworth economic game     189
The set of feasible allocations     190
The biloss operator     190
The Edgeworth box     192
Pareto minima     193
Core     193
Walras equilibria     194
Two-person zero-sum games     195
Duality gap and value     195
Saddle point     197
Perturbation by linear functions      198
Case of finite strategy sets: Matrix games     200
Two-Person Zero-Sum Games: Existence Theorems     204
The fundamental existence theorems     206
Existence of conservative solutions     208
Decision rules     211
Finite topology on convex subsets     211
Existence of an optimal decision rule     212
The Ky-Fan inequality     213
The Lasry theorem     214
The minisup theorem     216
The Nikaido theorem     217
Existence of saddle points     218
Another existence theorem for saddle points     218
Extension of games without and with exchange of informations     219
Definition of extensions of games     220
Mixed extensions     222
Extensions without exchange of information     223
Sequential extensions     225
Extensions with exchange of information     227
Iterated games     230
Iterated extensions     231
The Moulin theorem     233
Proof of playability of iterated extensions     233
A system of functional equations     236
A lemma on successive approximations     239
Proof of existence of saddle decision rules     240
The Fundamental Economic Model: Walras Equilibria     241
Description of the model     242
The subset of available commodities     242
Appropriation of the economy     244
Demand correspondences     244
Walras equilibrium     245
Examples of subsets of available commodities and of appropriations     245
Example: Quadratic demand functions     247
Existence of a Walras equilibrium     248
Existence of a Walras pre-equilibrium     248
Surjectivity of correspondences: the Debreu-Gale-Nikaido theorem     250
Demand correspondences defined by loss functions     251
Statement of the existence theorem     251
Upper semi-continuity of the demand correspondence     253
Compactification of an economy     254
Proof of the existence of a Walras equilibrium     256
Economies with producers     257
Description of the model     257
Statement of the existence theorem     258
Compactification     259
Proof of the existence of a Walras equilibrium     262
Non-Cooperative n-Person Games     263
Existence of a non-cooperative equilibrium      264
Games described in strategic form     264
Conservative values and multistrategies     265
Non-cooperative equilibria     266
The Nash theorem     267
Stability     268
Associated variational inequalities     269
Case of quadratic loss functions; application to Walras-Cournot equilibria     270
Non-cooperative games with quadratic loss functions     271
Existence of solutions of variational inequalities     272
Examples     274
Multistrategy sets defined by linear constraints     274
Walras-Cournot equilibria     276
Constrained non-cooperative games and fixed point theorems     279
Selection of a fixed point     279
Equilibria of constrained non-cooperative games     282
Fixed-point theorems     283
Non-cooperative Walras equilibria     285
Description of the model     285
Existence of a non-cooperative Walras equilibrium: the Arrow-Debreu theorem     286
Non-cooperative Walras equilibria of economies with producers     289
Main Solution Concepts of Cooperative Games     293
Behavior of the whole set of players: Pareto strategies     295
Pareto strategies      295
Rates of transfer     297
Pareto multipliers     297
Pareto allocations     300
Selection of Pareto strategies and imputations     303
Normalized games     304
Pareto strategies obtained by using selection functions     305
Closest strategy to the shadow minimum     306
The best compromise     307
Existence of Pareto strategies     308
Interpretation: threat functionals     308
Imputations: the Nash bargaining solution     309
Behavior of coalitions of players: the core     310
Coalitions     311
Cooperative game described in strategic form and its core     312
The multiloss operator F[superscript A]# of the coalition A     313
Examples of multistrategy sets X(A)     313
Economic games and core of an economy     314
Cooperative game described in characteristic form and its core     314
Behavior of fuzzy coalitions: the fuzzy core     316
Fuzzy coalitions     316
Extension of a family of coalitions     317
Debreu-Scarf coalitions     318
Fuzzy coalitions on a continuum of players     319
Fuzzy games described in characteristic form      320
Characterization of the core of a (fuzzy) game     320
Fuzzy economic games and fuzzy core of an economy     321
Fuzzy games described in strategic form and fuzzy core     324
Selection of elements of the core: cooperative equilibrium and nucleolus     329
Canonical cooperative equilibrium     329
Least-core     331
Nucleolus     333
Games With Side-Payments     336
Core of a fuzzy game with side-payments     338
Core of a game with side-payments     338
Linear games     340
Non-emptiness of the core of fuzzy games with side-payments     341
Core of fuzzy market games     343
Core of a game with side-payments     344
Convex cover of a game     345
Non-emptiness of the core of a balanced game     346
Balanced family of multistrategy sets     347
Balanced characteristic functions and convex loss functions     348
Further properties of convex functions and balances     351
Values of fuzzy games     353
The diagonal property     354
Sequence of fuzzy values     355
Existence and uniqueness of a sequence of fuzzy values     356
Relations between core and fuzzy value     359
Best approximation property of fuzzy values     358
Generalized solution to locally Lipschitz games     359
Shapley value and nucleolus of games with side-payments     360
The Shapley value     360
Existence and uniqueness of a Shapley value     361
Simple games     367
Nucleolus of games with side-payments     367
Games Without Side-Payments     370
Equivalence between the fuzzy core and the set of equilibria     370
Representation of a game     371
Equilibrium of a representation     373
Cover associated with a representation     374
Fuzzy core of a representation     376
The equivalence theorem     376
Non-emptiness of the fuzzy core of a balanced game     378
Statement of theorems of non-emptiness of the fuzzy core     379
Upper semi-continuity of the associated side-payment games     382
Existence of approximate cooperative equilibria     384
Proof of the non-emptiness of the core     386
Equivalence between the fuzzy core of an economy and the set of Walras allocations     386
Representation of economic games     386
Fuzzy core and Walras allocations      389
The equivalence theorem     390
Non-Linear Analysis and Optimal Control Theory     391
Minimax Type Inequalities, Monotone Correspondences and [gamma]-Convex Functions     393
Relaxation of compactness assumptions     395
Existence of a conservative solution     395
Proof of existence of a conservative solution     397
Existence of optimal decision rules and minisup under weaker compactness assumptions     399
Relaxation of continuity assumptions: variational inequalities for monotone correspondences     405
Variational inequalities     406
Existence of a solution to variational inequalities for completely upper semi-continuous correspondences     408
Pseudo-monotone functions: the Brezis-Nirenberg-Stampacchia theorem     410
Existence of a solution to variational inequalities for pseudo-monotone maps     413
Pseudo-monotonicity of monotone maps     414
Monotone and cyclically monotone correspondences     416
Maximal monotone correspondences     417
Relaxation of convexity assumptions     423
Definition of [gamma]-convex functions     424
The fundamental characteristic property of families of [gamma]-convex functions     424
The minisup theorem for [gamma subscript x]-convex-[gamma subscript y]-concave functions     426
Existence of optimal decision rules for functions [gamma subscript y]-concave with respect to y     428
Example: Image of a cone of convex functions by [pi]*     429
Relations between convexity and [gamma]-convexity     431
Example: [beta]-convex set functions     434
Example: Convex functions of atomless vector measures     436
Introduction to Calculus of Variations and Optimal Control     438
Duality in infinite dimensional spaces     441
Lagrangian of a minimization problem under linear constraints     443
Extremality relations     446
Existence of a Lagrange multiplier under the Slater condition     447
Relaxation of the Slater condition     449
Generalized Lagrangian of a minimization problem     451
Characterization of a Lagrangian by perturbations of the minimization problem     456
Duality in the case of non-convex integral criterion and contraints     458
Modulus of non-convexity of a function     459
Estimate of the duality gap     461
The Shapley-Folkman theorem     463
Sharp estimate of the duality gap     465
Applications     468
Extremality relations      470
The Aumann-Perles duality theorem     472
The approximation procedure     474
Duality in calculus of variations     476
The Green formula     480
Abstract problem of calculus of variations     482
The Hamiltonian system     484
Lagrangian of a problem of calculus of variations     486
Existence of a Lagrange multiplier     487
Example: the Dirichlet variational problem     488
The maximum principle for optimal control problems     492
Optimal control and impulsive control problems     497
The Hamilton-Jacobi-Bellman equation of a control problem     498
Construction of the closed loop control     502
The principle of optimality     503
The quadratic case: Riccati equations     505
The Bensoussan-Lions variational inequalities of a stopping time problem     508
Construction of the optimal stopping time     511
The Bensoussan-Lions quasi-variational inequalities of an impulsive control problem     511
Construction of the optimal impulsive control     515
Fixed Point Theorems, Quasi-Variational Inequalities and Correspondences     518
Fixed point and surjectivity theorems for correspondences     518
The Browder-Ky-Fan existence theorem for critical points     519
Properties of inward and outward correspondences     527
Critical points of homotopic correspondences     530
Other existence theorems for critical points     532
Quasi-variational inequalities     534
Selection of fixe